The Motion of a Material Point in a Newtonian Field under the Action of Equiangular Thrust

The problems of investigation and optimization of the motion of spacecraft are extensively discussed in the literature. Nevertheless, in many cases a large variety of qualitative characteristics of their motion and of the form of their trajectories are still unclear. In this paper we consider a plane equiangular acceleration of a spacecraft both in a Newtonian field and in its absence (at a large distance from the center of attraction). The general equation of a trajectory of plane acceleration is presented with the introduction of a new variable, an index of an exponent, which allows one to obtain convenient solutions at different values of the time-independent angle of inclination of the vector of thrust to the spacecraft's radius vector (i.e., when equiangular acceleration takes place). Asymptotic solutions are constructed and an interesting fact is revealed. Namely, it is shown that when the center of attraction exists or is absent, for all initial conditions the trajectories appearing at the above equiangular acceleration of a material point tend to the standard logarithmic spirals at a large distance from the center. Specifically, when the value of transverse (perpendicular to the radius vector) thrust is constant, there appears a logarithmic spiral with an angle of inclination to the radius vector equal to 35.264°. Different forms of the trajectory of equiangular acceleration of spacecraft at a low thrust are also studied. The results obtained can be useful for the investigation and choice of optimum space trajectories.     

从地球到日-火系平动点轨道的转移

平动点轨道特殊的空间位置及动力学特征,使其在深空探测中具有重要的应用。以日-火系平动点轨道(Lissajous与Halo轨道)任务为目标,结合平动点轨道的不变流形理论,研究了小推力转移问题。首先给出了圆型限制性三体动力学模型下平动点附近不变流形(稳定和不稳定流形)高阶分析解以及相应的计算实例。接着以流形分析解为基础,建立了初始小推力轨道优化模型,并利用改进的协作进化算法求解初始小推力轨道。最后将初始轨道离散,采用多点打靶法将最优控制问题转化为参数优化问题,并用序列二次规划方法(SQP)求解。仿真结果证明轨道设计方法的有效性。     

The use of solutions to problems of spacecraft trajectory optimization in impulse formulation when solving the problems of optimal control of trajectories of a spacecraft with limited thrust engine: I

In this first part of our paper, it is suggested to use solutions to boundary value problems in the optimization problems (in impulse formulation) for spacecraft trajectories in order to obtain the initial approximation, when boundary value problems of the maximum principle are solved numerically by the shooting method. The technique suggested is applied to the problems of optimal control over motion of the center of mass of a spacecraft controlled by the thrust vector of jet engine with limited thrust in an arbitrary gravitational field in a vacuum. The method is based on a modified (in comparison to the classic scheme) shooting method computation together with the method of continuation along a parameter (maximum reactive acceleration, initial thrust-to-weight ratio, or any other parameter equivalent to them). This technique allows one to obtain the initial approximation with a high precision, and it is applicable to a wide range of optimal control problems solved using the maximum principle, if the impulse formulation makes sense for these problems.     

变推力月球软着陆制导优化研究

提出了一种新的使用变推力火箭发动机实现月球定点软着陆制导的优化方法.在软着陆加速度抛物线(即二次函数)变化的条件下,通过一组代数方程连接初始条件和终端条件,避免了求解两点边值问题的迭代计算.给出了瞬时位置速度状态参数以及需要推力加速度、推力和秒流量的计算公式,并通过调整总飞行时间和着陆点位置实现了燃料消耗最小的优化处理.算例结果证明了这种方法的可行性和有效性.     

一种基于单站仅测角观测的弹道射向估计算法

针对单站仅测角条件下的弹道射向估计问题,提出了一种新的基于推力加速度模板的弹道射向估计算法。首先,建立以射向为待估参数的弹道平面切割模型,获得一组备选弹道曲线。其次,通过UKF算法从每条备选弹道曲线中估计出推力加速度曲线。最后,将得到的推力加速度曲线与推力加速度模板进行匹配,进而得到弹道射向估计结果。仿真结果验证了该算法的有效性。     

General solution for the optimal trajectory of an AFE-type spacecraft

The aeroassisted flight experiment (AFE) refers to an experimental spacecraft to be launched and then recovered by the Space Shuttle. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). In this paper, with reference to an AFE-type spacecraft, an actual GEO-to-LEO transfer is considered under the following assumptions: the GEO and LEO orbital planes are identical; both the initial and final orbits are circular; the initial phase angle is given, while the final phase angle is free. The aeroassisted orbital transfer trajectory involves three branches: a preatmospheric branch, GEO-to-entry; an atmospheric branch, entry-to-exit; a post-atmospheric branch, exit-to-LEO. The optimal trajectory is determined by minimizing the total characteristic velocity. The optimization is performed with respect to the velocity impulses at GEO, LEO, and the time history of the angle of bank during the atmospheric pass. It is assumed that the entry path inclination is free and that the angle of attack is constant, = 17.0 deg. The sequential gradient-restoration algorithm is used to compute the optimal trajectory and it is shown that the best atmospheric pass is to be performed with constant angle of bank. The resulting optimal trajectory constitutes an ideal nominal trajectory for the generation of guidance trajectories for two reasons: the fact that the low value of the characteristic velocity is accompanied by relatively low values of the peak heating rate and the peak dynamic pressure; and the simplicity of the control distribution, requiring constant angle of bank.     

形成三星星座的小推力变轨的时间最短控制

在研究和发展星座技术中，星座的发射是一项关键技术。本文针对形成三星星座，利用最优控制中的极小值原理，解算了用恒值、连接工作、牛顿级小推力变轨的时间最短控制问题。文中建立了最优小推力变轨的数学模型，求得了最优变轨的解析解，并通过牛顿下山法求解了三星星座变轨的小推力工作最优时间、最优方向和最优变轨轨迹。最后对星座变轨小推力最优控制工程实现的途径进行了探讨。为工程应用和研究提供参考。     

Experimental and theoretical modeling of laser propulsion

Some schemes of laser propulsive systems are discussed. The question concerned with a body acceleration due to series of air blast waves generated by laser sparks is studied. For this purpose the numerical solutions of gasdynamic equations are found under appropriate initial conditions corresponding to the real ones. Radiative losses and spatial effects at the nozzle exit are taken into account. Theoretical results presented as coupling coefficients (equivalent to reciprocal thrust cost realizing under periodical pulse laser operation) are compared with the experiment. Using conical and parabolic nozzles irradiated by pulsed CO₂ laser the thrust cost about 2000 W/N is achieved which is close to the minimum possible one for the air blast wave-nozzle wall interaction. The main characteristics of laser propulsive jet are presented. Experimental results on recoil momentum transfered to solids under their evaporation by the pulsed CO₂-laser are presented as well. The question of plasma shielding effects on the momentum transfer under the vapour optical breakdown conditions is touched on.     

轨道机动过程中推力加速度的实时最小方差估计

飞行器轨道机动过程中，为跟踪、定位机动目标和干预机动控制过程，需要统计处理离散的雷达观测量实时估计推进发动机的推力，进而确定飞行器的瞬时轨道参数。本文所述算法是该工程问题的探讨和解决方案。文章建立了轨道机动过程中连续变质量运动模型和离散雷达量测模型，推进发动机的质量秒耗量作为表征推力加速度的一个近似常量，应用扩展卡尔曼滤波对离散的雷达测量数据进行顺序统计处理给出秒耗量的最小方差估计；文章详细地推导了线性化量模型的变分方程和观测矩阵；仿真结果表明该算法能快速、准确地估计推进发动机的质量秒耗量和向机动目标施加的实际推力。     

远程导引脉冲变轨方案的有限推力修正

对远程导引脉冲变轨方案的有限推力修正进行了研究。将多脉冲变轨方案设计结果转换为有限推力式并进行修正,以消除转换误差和摄动模型误差。分析了由转换算法和摄动模型产生的初值误差,比较了不同工况下的修正效果。算例表明:将初始多脉冲变轨方案转换为有限推力后进行修正,所得结果能实现真实飞行环境中追踪航天器对目标航天器的拦截或交会。     

The use of solutions to problems of spacecraft trajectory optimization in impulse formulation when solving the problems of optimal control of trajectories of a spacecraft with limited thrust engine: II

As examples of application of the technique suggested in the first part of this work, the problems of optimizing the trajectories of spacecraft transfers between circular coplanar orbits are considered in this second part. During the transfer the spacecraft is controlled by the vector of thrust of a limited-thrust jet engine. The mass consumption is minimized for a limited time of transfer. Extreme trajectories with two and three powered sections (Homan-type and bi-elliptic transfer trajectories) are numerically determined. The solution of these well-studied problems allows one to compare the results of applying the suggested technique with the results of application of other previously used techniques.     

动力学补偿受控卫星轨道确定算法(英文)

受控卫星动力学模型中推力加速度的量级远远高于其他摄动的误差量级,观测量主要反映受控卫星动力学模型的误差。本文以跟踪和精确定位空间机动目标为目的,给出基于地面雷达观测,实时估计推力加速度,修正卫星动力学模型的轨道确定算法。通过建立连续推力控制过程变质量动力学模型,给出常推力变加速度满足的运动学微分方程; 建立变加速度估计系统状态方程,和扩展卡尔曼滤波轨道确定算法; 并给出连续推力控制卫星运动状态关于推力加速度的变分运动方程; 实际飞行控制应用表明: 利用地面测量数据,实时估计推力加速度并补偿系统动力学模型,解决了连续受控卫星轨道精确确定问题。     

基于推力加速度模板的主动段弹道跟踪方法

对弹道导弹的主动段跟踪是导弹防御中重要的环节。针对此问题提出了一种基于推力加速度模板的主动段跟踪算法，该算法需要主动段推力加速度信息支持。给出了一种简捷的推力加速度模板构建方法。利用近常速运动模型描述了导弹运动特性，解决了对导弹发射时间的估计问题和模板概率的递推计算问题，并将该方法应用到空间预警系统中，通过仿真验证了该方法具有较好的识别效果，位置和速度估计误差都接近于CRLB限。     

采用Gauss伪谱法的小推力日-火Halo轨道转移优化设计

针对日-地Halo轨道到日-火Halo轨道的小推力轨道转移问题,给出一种基于不变流形理论和Gauss伪谱法的优化设计方法。首先,在日心惯性坐标系中建立小推力轨道优化模型,并基于不变流形理论给出轨道转移中流形出口和入口的选择原则,应用该原则在日-地系统中选择流形出口,在日-火系统中选择流形入口,并将其作为轨道转移的初末状态;然后基于Gauss伪谱法将最优控制问题离散化为非线性规划(NLP)问题,并采用基于逆多项式的形状算法给出了NLP初值的计算方法;最后对该轨道转移问题进行了数学仿真。仿真结果表明：Gauss伪谱法可有效用于小推力日-火Halo轨道转移的优化,且采用逆多项式形状算法得到的初值具有初始误差小,使得NLP收敛速度快的特点。     

Optimization of transfer trajectories to the Apophis asteroid for spacecraft with high and low thrust

The problem of optimization of a spacecraft transfer to the Apophis asteroid is investigated. The scheme of transfer under analysis includes a geocentric stage of boosting the spacecraft with high thrust, a heliocentric stage of control by a low thrust engine, and a stage of deceleration with injection to an orbit of the asteroid’s satellite. In doing this, the problem of optimal control is solved for cases of ideal and piecewise-constant low thrust, and the optimal magnitude and direction of spacecraft’s hyperbolic velocity “at infinity” during departure from the Earth are determined. The spacecraft trajectories are found based on a specially developed comprehensive method of optimization. This method combines the method of dynamic programming at the first stage of analysis and the Pontryagin maximum principle at the concluding stage, together with the parameter continuation method. The estimates are obtained for the spacecraft’s final mass and for the payload mass that can be delivered to the asteroid using the Soyuz-Fregat carrier launcher.     

电动帆航天器谷神星探测任务轨迹优化

针对电动帆航天器谷神星探测任务轨迹优化问题,提出一种基于高斯伪谱法和遗传算法的混合优化算法。为了验证所提出的混合优化算法有效性,并考察任务起始时间和电动帆特征加速度对探测任务的影响,进行了一定数量的数值仿真。仿真结果表明：电动帆航天器自地球至谷神星的飞行时间随着起始时间的变化呈周期性波动,波动周期基本与地球和谷神星的会合周期一致;电动帆航天器的特征加速度越小,完成过渡所需要的飞行时间越长,且一个具有中等特征加速度的电动帆航天器便能在可接受的时间内完成自地球至谷神星的过渡;所提出的混合优化算法是有效的,能够在无任何初值猜测的情况下完成电动帆航天器飞行轨迹的优化。     

Optimal flights to near-Earth asteroid

The spacecraft flights to the Near-Earth asteroid in order to give an impact influence on the asteroid, correct its orbit and prevent the asteroid’s collision with the Earth are analyzed.In the first part, the impulse flights are analyzed in the Lambert approach. There are determined the optimal trajectories maximizing the asteroid deviation from the Earth.In the second part, the flights with the chemical and electric-jet engines are analyzed. The high thrust is used to launch the spacecraft from the geocentric orbit, and the low thrust is applied for the heliocentric motion. On the base of optimal impulse transfer, the optimal low thrust trajectories are determined using Pontryagin maximum principle.The numerical results are given for the flight to the asteroid Toutatis. Parameters of the spacecraft impact on the asteroid are determined. The asteroid deviation from the Earth caused by the spacecraft influence is presented.     

基于退火遗传算法的小推力轨道优化问题研究

利用退火遗传算法解决小推力轨道优化问题。首先利用传统混合法将轨道优化问题归结为受非线性方程约束的参数优化问题。通过结合退火和随机惩罚函数对约束条件进行处理后，用遗传算法求解这个参数优化问题。最后再采用局部优化算法提高解的精度。这种算法既保持了传统混合法精度高、解轨线光滑的优点，又克服了传统轨道优化方法收敛性差、初始猜测困难、容易陷入局部极小解的缺点。在本文的最后，利用文中提出的轨道优化算法求解“喷-停-喷”型定常推力幅值地球-木星轨道转移问题。算例证明此算法可以有效地求解小推力轨道转移问题，尤其适用于传统轨道优化方法难以求解的复杂轨道优化问题。     

Constant-thrust glideslope guidance algorithm for time-fixed rendezvous in real halo orbit

This paper presents a fixed-time glideslope guidance algorithm that is capable of guiding the spacecraft approaching a target vehicle on a quasi-periodic halo orbit in real Earth–Moon system. To guarantee the flight time is fixed, a novel strategy for designing the parameters of the algorithm is given. Based on the numerical solution of the linearized relative dynamics of the Restricted Three-Body Problem (expressed in inertial coordinates with a time-variant nature), the proposed algorithm breaks down the whole rendezvous trajectory into several arcs. For each arc, a two-impulse transfer is employed to obtain the velocity increment (delta-v) at the joint between arcs. Here we respect the fact that instantaneous delta-v cannot be implemented by any real engine, since the thrust magnitude is always finite. To diminish its effect on the control, a thrust duration as well as a thrust direction are translated from the delta-v in the context of a constant thrust engine (the most robust type in real applications). Furthermore, the ignition and cutoff delays of the thruster are considered as well. With this high-fidelity thrust model, the relative state is then propagated to the next arc by numerical integration using a complete Solar System model. In the end, final corrective control is applied to insure the rendezvous velocity accuracy. To fully validate the proposed guidance algorithm, Monte Carlo simulation is done by incorporating the navigational error and the thrust direction error. Results show that our algorithm can effectively maintain control over the time-fixed rendezvous transfer, with satisfactory final position and velocity accuracies for the near-range guided phase.     

Novel orbits of Mercury,Venus and Mars enabled using low-thrust propulsion

Exploration of the inner planets of the Solar System is vital to significantly enhance the understanding of the formulation of the Earth and other planets. This paper therefore considers the development of novel orbits of Mars, Mercury and Venus to enhance the opportunities for remote sensing of these planets. Continuous acceleration is used to extend the critical inclination of highly elliptical orbits at each planet and is shown to require modest thrust magnitudes. This paper also presents the extension of existing sun-synchronous orbits around Mars. However, unlike Earth and Mars, natural sun-synchronous orbits do not exist at Mercury or Venus. This research therefore also uses continuous acceleration to enable circular and elliptical sun-synchronous orbits, by ensuring that the orbit's nodal precession rate matches the planets mean orbital rate around the Sun, such that the lighting along the ground-track remains approximately constant over the mission duration. This property is useful both in terms of spacecraft design, due to the constant thermal conditions, and for comparison of images. Considerably high thrust levels are however required to enable these orbits, which are prohibitively high for orbits with inclinations around 90°. These orbits therefore require some development in electric propulsion systems before becoming feasible.